Null controllability of one-dimensional quasilinear parabolic equations via multiplicative controls

This paper is concerned with the null controllability problem for a class of quasilinear parabolic equations under multiplicative control, locally supported in space. For the purpose of proving the existence of a multiplicative control forcing the solution rest at a time $T>0$, we need to establish the decay property of solutions for the system without control first. We have obtained decay estimates for the $L^\infty$-norm and the $H^1$-norm of solutions to the homogenous quasilinear parabolic equations. Notably, the decay of the $L^\infty$-norm requires no smallness condition on the initial data, whereas the decay of the $H^1$-norm requires that the $L^\infty$-norm remains small. Based on the decay estimates and maximum modulus estimate of solutions to quasilinear parabolic equations, together with the local null controllability of quasilinear parabolic equations under additive controls, we prove the null controllability of the quasilinear parabolic equations via multiplicative controls. As a byproduct, we also obtain the global null controllability for large time to the quasilinear parabolic equations via additive controls. Given that the controllability under multiplicative control is achieved over a long time horizon, we finally investigate the existence of time optimal control.